TAILIEUCHUNG - Đề tài " Classification of prime 3manifolds with σ-invariant greater than RP3 "

In this paper we compute the σ-invariants (sometimes also called the smooth Yamabe invariants) of RP3 and RP2 × S 1 (which are equal) and show that the only prime 3-manifolds with larger σ-invariants are S 3 , S 2 × S 1 , and ˜ S 2 ×S 1 (the nonorientable S 2 bundle over S 1 ). More generally, we show that any 3-manifold with σ-invariant greater than RP3 is either S 3 , a connect sum with an S 2 bundle over S 1 , or has more than one nonorientable prime component. A corollary. | Annals of Mathematics Classification of prime 3-manifolds with Q-invariant greater than RP3 By Hubert L. Bray and Andr re Neves Annals of Mathematics 159 2004 407-424 Classification of prime 3-manifolds with -invariant greater than RP3 By Hubert L. BRAy and André Neves Abstract In this paper we compute the ơ-invariants sometimes also called the smooth Yamabe invariants of RP3 and RP2 X S1 which are equal and show that the only prime 3-manifolds with larger ơ-invariants are S3 S2 X S1 and S2XS1 the nonorientable S2 bundle over S1 . More generally we show that any 3-manifold with ơ-invariant greater than RP3 is either S3 a connect sum with an S2 bundle over S1 or has more than one nonorientable prime component. A corollary is the Poincare conjecture for 3-manifolds with ơ-invariant greater than RP3. Surprisingly these results follow from the same inverse mean curvature flow techniques which were used by Huisken and Ilmanen in 7 to prove the Riemannian Penrose Inequality for a black hole in a spacetime. Richard Schoen made the observation 18 that since the constant curvature metric which is extremal for the Yamabe problem on RP3 is in the same conformal class as the Schwarzschild metric which is extremal for the Penrose inequality on RP3 minus a point there might be a connection between the two problems. The authors found a strong connection via inverse mean curvature flow. 1. Introduction We begin by reminding the reader of the definition of the ơ-invariant of a closed 3-manifold and some of the previously known results. Since our results only apply to 3-manifolds we restrict our attention to this case. Given a closed 3-manifold M the Einstein-Hilbert energy functional on the space of metrics g is defined to be the total integral of the scalar curvature The research of the first author was supported in part by NSF grant DMS-0206483. The research of the second author was kindly supported by FCT-Portugal grant BD 893 2000. 408 HUBERT L. BRAY AND ANDRE NEVES Rg after .

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